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What you will learn in Quantum Programming in Depth

Solving problems with Q# and Qiskit

by Mariia Mykhailova

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Chapter by chapter

Concepts explained in this book

• quantum oracleA marking oracle is a quantum oracle that identifies selected computational-basis states by applying a designated marking transformation to them.
• phase kickbackPhase kickback is a quantum effect in which a controlled operation transfers a phase associated with its target to the relative phase of the control-qubit state.
• reversible operationA reversible function is a function in which every input has exactly one output and every output is produced by exactly one input.
• quantum superpositionSuperposition is a quantum state formed as a coherent linear combination of multiple basis states with complex amplitudes.
• state preparationA state preparation algorithm is a computational procedure that transforms an initial quantum state into a specified target quantum state.
• controlled quantum operationA controlled gate is a quantum gate whose operation on a target system is conditioned on one or more control qubits.
• eigenphaseAn eigenphase is the phase associated with an eigenvalue of a unitary operator, typically expressed as the angle of that eigenvalue on the complex unit circle.
• classical functionA mapping from classical inputs to classical outputs that specifies the function implemented by an oracle.
• resource estimationResource estimation is analysis that estimates the computational resources required by a program or algorithm, such as qubits, gates, runtime, or hardware capacity.
• n-queens problemThe N queens puzzle is the problem of placing N chess queens on an N-by-N board so that no two queens attack each other.
• unitary transformationA unitary transformation is a linear transformation that preserves inner products and therefore quantum-state norm and measurement probabilities.
• QiskitQiskit is an open-source software development kit for writing and executing quantum programs, including a Python library interface.
• Grover's algorithmGrover's search algorithm is a quantum algorithm that finds a marked item in an unstructured search space with quadratic query-speedup over classical exhaustive search.
• quantum operationA quantum operation is a transformation or physical process acting on a quantum state, including unitary gates and measurement.
• computational basisThe computational basis is the reference basis of quantum states labeled by binary strings, including |0⟩ and |1⟩ for one qubit.
• controlled-on-zero gateA controlled-on-zero gate applies its target operation when its control qubits are in the zero state.
• quantum phase estimationQuantum phase estimation is a quantum algorithm for estimating the eigenphase associated with an eigenvalue of a unitary operator, given a corresponding eigenstate.
• probability theoryProbability is a numerical value between zero and one expressing the likelihood of a measurement outcome.
• quantum simulatorA quantum simulator is a classical computer program that simulates the behavior of a quantum system or quantum program.
• resultA measurement outcome is the classical result produced when a quantum measurement is performed on a quantum state.
• search spaceThe set of candidate inputs or basis states over which a search problem ranges.
• Q#Q# is Microsoft's domain-specific programming language for expressing quantum algorithms and operations.
• iterative phase estimationAdaptive phase estimation is a quantum algorithm that estimates an eigenphase incrementally by adjusting later circuit operations based on earlier measurement results.
• eigenvectorAn eigenstate is a quantum state that a quantum gate transforms into the same state multiplied by a scalar phase or eigenvalue.
• little-endianLittle-endian encoding orders a bit string or composite value with its least significant bit or component first.
• quantum measurementA quantum measurement is an observation process that maps a quantum state to a classical outcome according to quantum probabilities.
• quantum computingQuantum computing is computation that uses quantum-mechanical states and operations to process information.
• ancillary qubitAn auxiliary qubit is an additional qubit allocated to hold intermediate information during a quantum computation.
• logical NOTNegation is the logical operation that changes a true value to false and a false value to true.
• quantum computerQuantum hardware is physical computing equipment that stores and manipulates information in quantum states.
• uncomputationUncomputation is the reversible execution of a computation's inverse to erase intermediate information while preserving the desired output.
• sparse simulationA sparse simulator is a quantum simulator that represents and manipulates only a small subset of the basis states in a quantum state.
• diagonal matrixA diagonal matrix is a square matrix whose entries outside the main diagonal are zero.
• reflection about the meanReflection about the mean is a quantum operation that reflects amplitude values about their average amplitude.
• rotational gateAn Ry gate is a single-qubit rotation gate that rotates a qubit state around the Y axis of the Bloch sphere.
• big-endianBig-endian encoding orders a bit string or composite value with its most significant bit or component first.
• binary stringA bit string is a finite sequence of binary digits used here to represent a computational basis state.
• error correctionError correction is a technique that detects and corrects computational errors so that encoded logical information remains reliable despite hardware noise.
• truth tableA truth table is a tabular representation of the output of a Boolean operation for every possible combination of binary inputs.
• quantum algorithmA quantum algorithm is a computational procedure designed to run on a quantum computer and exploit quantum-mechanical operations or information.
• testA test case is a specified input scenario and expected result used to evaluate software behavior.
• hybrid quantum-classical computingA hybrid quantum-classical structure is a computational organization that combines classical processing with quantum computation and subsequent classical processing.
• x gateAn X gate is a single-qubit bit-flip quantum gate that exchanges the computational-basis states zero and one.
• quantum supremacyA claim that a quantum device has solved a problem that no classical computer can solve in a reasonable amount of time.
• Azure Quantum Resource EstimatorThe Azure Quantum Resource Estimator is a Microsoft tool that estimates the resources required to execute quantum programs on specified fault-tolerant quantum-computing architectures.
• search problemThe problem of finding an input value that satisfies a specified condition, such as a value for which a function returns one.
• phase gateThe Z gate is the single-qubit Pauli gate that leaves ket zero unchanged and multiplies ket one by negative one.
• Boolean clauseA Boolean clause is a component of a Boolean formula formed by a disjunction of literals.
• parityParity is the property of being even or odd, here determined for a computational-basis state by the parity of its bit values.
• cosine-sine factorA cosine-sine matrix, or CS matrix, is the structured matrix in a cosine-sine decomposition whose blocks contain cosine and sine values.
• unitary gate synthesisUnitary gate synthesis is the construction of a quantum-gate sequence that implements a specified unitary transformation, possibly approximately.
• unstructured searchA search problem in which the possible inputs have no exploitable structure for organizing or accelerating lookup.
• nonorthogonal stateA nonorthogonal state is a quantum state having a nonzero inner product with another state, so the two states cannot be perfectly distinguished by measurement.
• logical ORThe OR operator returns true when at least one of its Boolean operands is true.
• quadratic speedupA quadratic speedup is an algorithmic performance improvement in which runtime scales as the square root of the corresponding classical runtime.
• logical architectureA logical architecture is the organization of error-corrected quantum computation, including its error-correction code, logical-qubit connectivity, and supported logical operations.
• magic gateA magic gate is a fault-tolerant quantum operation that is not directly supported by a logical architecture and is implemented using magic states.
• magic state distillationMagic state distillation is a fault-tolerant procedure that converts multiple noisy resource states into fewer higher-fidelity magic states for quantum computation.
• relative phaseRelative phase is the phase difference between quantum amplitudes that affects interference independently of their magnitudes.
• Grover iterateA Grover iteration is one repetition of Grover's search procedure, combining an oracle operation with a reflection about the mean.
• Bell stateA Bell pair is a maximally entangled two-qubit quantum state.
• quantum amplitudeA quantum amplitude is a complex coefficient of a basis state whose squared magnitude determines its measurement probability.
• nonreversible functionA nonreversible function is a function for which the mapping cannot be uniquely inverted, such as when multiple inputs correspond to one output.
• cosine-sine decompositionA cosine-sine decomposition is a matrix factorization that represents an even-dimensional unitary matrix as a product of two block-diagonal unitary matrices and a structured cosine-sine factor.
• post-processingClassical postprocessing is the processing of quantum-computation results using classical computation.
• De Morgan's lawsDe Morgan's laws are Boolean identities relating negated conjunctions and disjunctions.
• equality functionAn equality function outputs one exactly when its two binary inputs are equal.
• out-of-place implementationAn out-of-place implementation computes a result in additional storage while preserving the input values.
• global phaseA global phase is a common complex phase factor multiplying every component of a quantum state.
• Hadamard gateThe Hadamard gate is a single-qubit quantum gate that creates equal superpositions from computational-basis states.
• permutation matrixA permutation matrix is a binary square matrix with exactly one 1 in every row and column.
• QuantumCircuitQuantumCircuit is a Qiskit class representing a quantum circuit and exposing operations for constructing or executing it.
• meanThe mean is the equal-superposition quantum state over all basis states in a specified search space.
• non-solution stateA non-solution state is the normalized superposition of basis states in the search space that do not satisfy the problem condition.
• solution stateA solution state is the normalized superposition of basis states whose inputs satisfy the problem condition.
• custom quantum gateA custom quantum gate is a user-defined quantum operation implementing a specified reversible transformation.
• addition modulo twoAddition modulo two is binary addition in which results are reduced to their remainder modulo two.
• pytestpytest is a Python testing framework for discovering, organizing, and running automated tests.
• logical gateA logical gate is an error-corrected quantum operation acting on one or more logical qubits.
• magic stateA magic state is a specially prepared ancillary quantum state that enables fault-tolerant implementation of otherwise unsupported quantum gates.
• performance estimationPerformance estimation is the analysis of the time and computational resources required to execute an algorithm on a specified computational architecture.
• state vector simulatorA full-state simulator represents every amplitude of a quantum state vector when simulating a quantum program.
• Bennett’s trickBennett’s trick is a reversible computation technique that copies an output to a separate register and then runs the computation backward to reset intermediate qubits.
• quantum entanglementEntanglement is a quantum correlation in which a composite system cannot be represented as independent states of its component systems.
• quantum interferenceInterference is the alteration of quantum amplitudes through their constructive or destructive combination.
• column constraintA column constraint is the N queens condition that each board column contains exactly one queen.
• diagonal constraintA diagonal constraint is the N queens condition that each board diagonal contains at most one queen.
• oracle queryA query is one evaluation of a black-box function or oracle on a supplied input.
• quantum stateA quantum state is a mathematical representation of the physical condition of a quantum system, including the probabilities of measurement outcomes.
• adjointThe adjoint of a complex matrix or linear operator is its conjugate transpose, and for a unitary operator it equals the inverse.
• least-significant bitThe least significant bit is the lowest-place bit in a binary representation and has the smallest positional value.
• amplitude encodingAmplitude encoding represents information in the absolute values of the amplitudes of quantum basis states.
• phase encodingPhase encoding represents information in the relative phases of the amplitudes of quantum basis states.
• Shor's algorithmShor's algorithm is a quantum algorithm for integer factorization with asymptotically better performance than the best known classical factorization algorithms.
• classical computerConventional computing based on classical physical systems and information representations rather than quantum-mechanical effects.
• binary representationBinary notation represents an integer or index as an ordered sequence of bits.
• classical preprocessingClassical preprocessing is the preparation and conversion of problem data before it is supplied to a quantum computation.
• inverse quantum Fourier transformThe inverse quantum Fourier transform is the inverse unitary transformation of the quantum Fourier transform, mapping encoded phase information back toward computational-basis information.
• Q# operationA Q# operation is a callable Q# construct that can accept parameters, act on qubits, and return a value.
• constraintA constraint is a condition that a candidate queen placement must satisfy, such as prohibiting two queens from sharing a row, column, or diagonal.
• cell-based board encodingA cell-based board encoding represents each cell of a chessboard with one binary bit indicating whether that cell contains a queen.
• quantum softwareSoftware that supports the development, testing, evaluation, execution, or operation of quantum-computing applications.
• measurement basisA measurement basis is a pair of orthogonal quantum states that defines the possible outcomes of a quantum measurement.
• input encodingAn input encoding is a representation that maps each queen's row position to the integer index of its occupied column.
• quantum programmingQuantum programming is the development of programs that specify or control computations on quantum systems.
• eigenvalueAn eigenvalue is the scalar, here a unit-modulus complex phase factor, by which a linear transformation scales an associated eigenvector.
• bitA bit is a binary digit that represents either zero or one.
• S† gateThe S† gate is the adjoint of the S phase gate and applies a phase of negative pi over two to the one computational-basis state relative to zero.
• joint measurementA joint measurement is a quantum measurement performed on multiple qubits or subsystems to obtain information about their collective properties or correlations.
• gate applicationThe programming operation of applying a quantum gate to one or more qubits.
• trapped ionA qubit encoded in the quantum state of an ion confined by electromagnetic fields.
• control patternA control pattern is a bit string or equivalent encoding specifying the basis values required on control qubits.
• orthogonalityAn orthogonal state is a quantum state whose inner product with another state is zero, allowing the two states to be perfectly distinguished by a suitable measurement.
• parity trickThe parity trick checks whether counts satisfy a constraint by computing their values modulo two instead of counting them exactly.
• W stateA W state is a multipartite quantum state that is an equal superposition of computational-basis states containing exactly one excited qubit.
• conjugationConjugation, or the Q# within-apply construct, performs a computation, applies an operation, and then automatically applies the computation's adjoint to uncompute intermediate state.
• unit typeUnit is a type with exactly one value, representing the absence of a meaningful return value.
• controlled-Z gateA controlled-Z gate is a two-qubit controlled phase gate that applies a phase flip when both qubits are in the one state.
• control qubitA control qubit is a qubit whose state determines whether a controlled gate operates on its target qubits.
• noisy intermediate-scale quantum eraThe stage of quantum-computing development characterized by devices with imperfect qubits that are not fully fault tolerant or broadly useful for computation.
• conjunctive normal formConjunctive normal form is a representation of a Boolean expression as a conjunction of clauses, with each clause formed from disjunctive literals.
• permutationA permutation is an arrangement containing each member of a finite set exactly once.
• error budgetAn error budget is the maximum allowed probability that an algorithm fails during execution.
• quantum state tomographyQuantum state tomography is a measurement-based technique for estimating an unknown quantum state from results collected from multiple identically prepared copies.
• block-diagonal unitaryA block-diagonal unitary is a unitary transformation represented by a block-diagonal matrix whose diagonal blocks are the active subtransformations.
• satisfiability problemThe Boolean satisfiability problem is the problem of finding an assignment to Boolean variables that makes a given Boolean expression true.
• quantum state discriminationState discrimination is the task of identifying which quantum state was provided from a specified set of possible states.
• MResetZMResetZ is a Q# operation that measures a qubit in the computational basis, resets it to the zero state, and returns a Result.
• qsharp.runqsharp.run is a Python API method that executes a Q# operation using a simulator instance.
• ResultResult is the Q# data type whose values represent quantum measurement outcomes as Zero or One.
• quantum system simulationQuantum system simulation is the computational modeling of the behavior and properties of physical systems governed by quantum mechanics.
• real amplitudeA real amplitude is a quantum amplitude whose imaginary component is zero.
• parity measurementA parity measurement is a quantum measurement that determines whether the combined bit values of a computational-basis state have even or odd parity.
• CNOT gateA CNOT gate is a two-qubit controlled-NOT gate that flips a target qubit when its control qubit is one.
• controlled-Ry gateA controlled-Ry gate is a controlled rotation gate that applies an Ry rotation to target qubits when its control qubits satisfy the control condition.
• target qubitA target qubit is a qubit on which a controlled gate conditionally performs its operation.
• technology stackA software stack is a layered collection of software components that collectively supports application execution on a computing platform.
• Azure Quantum Development KitThe Azure Quantum Development Kit is a Visual Studio Code extension for developing and executing Q# programs and accessing quantum-development actions.
• reset methodA reset operation measures a qubit and applies an X gate when necessary to return it to the computational-basis zero state.
• use statementA use statement is a Q# construct that allocates qubits for use within a program scope.
• parameterized gateA parameterized gate is a quantum gate whose transformation depends on one or more classical parameter values.
• quantum Fourier transformThe quantum Fourier transform is the quantum analogue of the discrete Fourier transform, mapping amplitudes between computational bases.
• quantum circuitA quantum circuit is a representation of a quantum computation as an ordered sequence of quantum gates and measurements.
• ApplyControlledOnIntApplyControlledOnInt is a Q# standard-library operation that applies a gate when control qubits match an integer-encoded bit pattern.
• Q# functionA Q# function is a deterministic classical callable in Q# that returns a value without producing side effects.
• error mitigationA family of techniques that reduces the impact of observed quantum-computation errors without fully correcting them during execution.
• fault-tolerant quantum computerA fault-tolerant quantum computer is a quantum computer that suppresses computational errors sufficiently to perform arbitrarily long computations reliably.
• neutral atomA quantum-computing platform that encodes and manipulates quantum information in electrically neutral atoms held in controlled configurations.
• photonA quantum-computing platform that encodes and processes information using individual particles of light.
• proof-of-concept deviceA device that demonstrates the fundamental feasibility of performing a computation with quantum-mechanical principles.
• superconducting circuitA quantum-computing platform whose quantum states are implemented with electrical circuits containing superconducting elements.

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